小q拉蓋爾多項式

4th order Little q-Laguerre polynomials

小q拉蓋爾多項式是一個以基本超幾何函數定義的正交多項式

$\displaystyle p_{n}(x;a|q)={}_{2}\phi _{1}(q^{-n},0;aq;q,qx)={\frac {1}{(a^{-1}q^{-n};q)_{n}}}{}_{2}\phi _{0}(q^{-n},x^{-1};;q,x/a)$

極限關係

大q拉蓋爾多項式→小q拉蓋爾多項式

在大q拉蓋爾多項式中，令$x\to bqx$,並令$b\to \infty$即得小q拉蓋爾多項式

$\lim _{b\to \infty }P_{n}(bqx;a,b;q)=p_{n}(x;a|q)$

仿射Q克拉夫楚克多項式→ 小q拉蓋爾多項式：

$\lim _{a\to 1}=K_{n}^{aff}(q^{x-N};p,N|q)=p_{n}(q^{x};p,q)$ 令小q拉蓋爾多項式 $a=q^{a}$ $x=(1-q)*x$,然後令q→1 即得拉蓋爾多項式

$\lim _{q\to 1}P_{a}(1-q)x;q^{a}|q)={\frac {L_{n}^{(a)}(x)}{L_{n}^{(a)}(0)}}$

驗證 9階小q拉蓋爾多項式→拉蓋爾多項式

作上述代換，

$P_{a}(1-q)x;q^{a}|q)=1+{\frac {qx}{1-{q}^{\alpha }q}}-{\frac {x}{{q}^{8}\left(1-{q}^{\alpha }q\right)}}$ $+\left(1-{q}^{-9}\right)\left(1-{q}^{-8}\right){q}^{2}\left(1-q\right){x}^{2}\left(1-{q}^{2}\right)^{-1}\left(1-{q}^{\alpha }q\right)^{-1}\left(1-{q}^{\alpha }{q}^{2}\right)^{-1}$$+\left(1-{q}^{-9}\right)\left(1-{q}^{-8}\right)\left(1-{q}^{-7}\right){q}^{3}\left(1-q\right)^{2}{x}^{3}\left(1-{q}^{2}\right)^{-1}$$\left(1-{q}^{3}\right)^{-1}\left(1-{q}^{\alpha }q\right)^{-1}\left(1-{q}^{\alpha }{q}^{2}\right)^{-1}\left(1-{q}^{\alpha }{q}^{3}\right)^{-1}+\cdots$

求q→1極限得

令a=3,得 $(1-{\frac {9}{4}}x+{\frac {9}{5}}{x}^{2}-{\frac {7}{10}}{x}^{3}+{\frac {3}{20}}{x}^{4}-{\frac {3}{160}}{x}^{5}+{\frac {1}{720}}{x}^{6}-{\frac {1}{16800}}{x}^{7}+{\frac {1}{739200}}{x}^{8}-{\frac {1}{79833600}}{x}^{9})$

另一方面

${\frac {L_{n}^{(3)}(x)}{L_{n}^{(3)}(0)}}$ =$(1-{\frac {9}{4}}x+{\frac {9}{5}}{x}^{2}-{\frac {7}{10}}{x}^{3}+{\frac {3}{20}}{x}^{4}-{\frac {3}{160}}{x}^{5}+{\frac {1}{720}}{x}^{6}-{\frac {1}{16800}}{x}^{7}+{\frac {1}{739200}}{x}^{8}-{\frac {1}{79833600}}{x}^{9})$

二者顯然相等 QED

圖集

LITTLE Q-LAGUERRE POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS IM COMPLEX 3D MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS RE COMPLEX 3D MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS ABS DENSITY MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS IM DENSITY MAPLE PLOT

LITTLE Q-LAGUERRE POLYNOMIALS RE DENSITY MAPLE PLOT

參考文獻

• Chihara, Theodore Seio, An introduction to orthogonal polynomials, Mathematics and its Applications 13, New York: Gordon and Breach Science Publishers, 1978, ISBN 978-0-677-04150-6, MR 0481884, Reprinted by Dover 2011 
• Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719 
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, 2010, ISBN 978-3-642-05013-8, MR 2656096, doi:10.1007/978-3-642-05014-5 
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Chapter 18: Orthogonal Polynomials, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248 
• Van Assche, Walter; Koornwinder, Tom H., Asymptotic behaviour for Wall polynomials and the addition formula for little q-Legendre polynomials, SIAM Journal on Mathematical Analysis, 1991, 22 (1): 302–311 [2024-06-17], ISSN 0036-1410, MR 1080161, doi:10.1137/0522019, （原始內容存檔於2024-07-21） 
• Wall, H. S., A continued fraction related to some partition formulas of Euler, The American Mathematical Monthly, 1941, 48 (2): 102–108, ISSN 0002-9890, JSTOR 2303599, MR 0003641, doi:10.1080/00029890.1941.11991074